


from __future__ import annotations

from typing import Generic, TypeVar

UpperCamelCase					:       Optional[Any]			    =     TypeVar("""T""")



class A__     ( Generic[T]  ):



  """simple docstring"""



  def __init__( self			:  Tuple				,				lowerCamelCase__			:  T       ):
        a__	:  int     =   data
        a__	:  List[Any]     =   self
        a__	:  Optional[Any]     =   0




class A__     ( Generic[T]  ):



  """simple docstring"""



  def __init__( self			:  Union[str, Any]       ):
        # map from node name to the node object
        a__	:  dict[T, DisjointSetTreeNode[T]]     =   {}

  def    _UpperCamelCase( self			:  List[Any]				,				lowerCamelCase__			:  T       ):
        # create a new set with x as its member
        a__	:  str     =   DisjointSetTreeNode(lowerCamelCase__       )

  def    _UpperCamelCase( self			:  int				,				lowerCamelCase__			:  T       ):
        # find the set x belongs to (with path-compression)
        a__	:  Optional[Any]     =   self.map[data]
        if elem_ref != elem_ref.parent:
              a__	:  Optional[int]     =   self.find_set(elem_ref.parent.data       )
        return elem_ref.parent

  def    _UpperCamelCase( self			:  Optional[int]				,				lowerCamelCase__			:  DisjointSetTreeNode[T]				,				lowerCamelCase__			:  DisjointSetTreeNode[T]       ):
        # helper function for union operation
        if nodea.rank > nodea.rank:
              a__	:  Tuple     =   nodea
        else:
              a__	:  Any     =   nodea
              if nodea.rank == nodea.rank:
                    nodea.rank += 1

  def    _UpperCamelCase( self			:  List[str]				,				lowerCamelCase__			:  T				,				lowerCamelCase__			:  T       ):
        # merge 2 disjoint sets
        self.link(self.find_set(lowerCamelCase__       )				,				self.find_set(lowerCamelCase__       )       )




class A__     ( Generic[T]  ):



  """simple docstring"""



  def __init__( self			:  List[Any]       ):
        # connections: map from the node to the neighbouring nodes (with weights)
        a__	:  dict[T, dict[T, int]]     =   {}

  def    _UpperCamelCase( self			:  Union[str, Any]				,				lowerCamelCase__			:  T       ):
        # add a node ONLY if its not present in the graph
        if node not in self.connections:
              a__	:  List[str]     =   {}

  def    _UpperCamelCase( self			:  Tuple				,				lowerCamelCase__			:  T				,				lowerCamelCase__			:  T				,				lowerCamelCase__			:  int       ):
        # add an edge with the given weight
        self.add_node(lowerCamelCase__       )
        self.add_node(lowerCamelCase__       )
        a__	:  Tuple     =   weight
        a__	:  Union[str, Any]     =   weight

  def    _UpperCamelCase( self			:  Union[str, Any]       ):
        a__	:  List[Any]     =   []
        a__	:  Dict     =   set()
        for start in self.connections:
              for end in self.connections[start]:
                    if (start, end) not in seen:
                          seen.add((end, start)       )
                          edges.append((start, end, self.connections[start][end])       )
        edges.sort(key=lambda lowerCamelCase__       : x[2]       )

        # creating the disjoint set
        a__	:  Optional[Any]     =   DisjointSetTree[T]()
        for node in self.connections:
              disjoint_set.make_set(lowerCamelCase__       )

        # MST generation
        a__	:  List[Any]     =   0
        a__	:  Union[str, Any]     =   0
        a__	:  Dict     =   GraphUndirectedWeighted[T]()
        while num_edges < len(self.connections       ) - 1:
              a__,	a__,	a__	:  str     =   edges[index]
              index += 1
              a__	:  str     =   disjoint_set.find_set(lowerCamelCase__       )
              a__	:  List[str]     =   disjoint_set.find_set(lowerCamelCase__       )
              if parent_u != parent_v:
                    num_edges += 1
                    graph.add_edge(lowerCamelCase__				,				lowerCamelCase__				,				lowerCamelCase__       )
                    disjoint_set.union(lowerCamelCase__				,				lowerCamelCase__       )
        return graph


"""simple docstring"""


from __future__ import annotations

from typing import Any






class   __UpperCAmelCase		:

				def __init__( self      ,					_lowerCamelCase      ,					_lowerCamelCase      ,					_lowerCamelCase = 0 ):
										lowerCamelCase__   , lowerCamelCase__   =row, column
										lowerCamelCase__   =[[default_value for c in range(_lowerCamelCase )] for r in range(_lowerCamelCase )]

				def __str__( self ):
										lowerCamelCase__   =F'''Matrix consist of {self.row} rows and {self.column} columns\n'''

										# Make string identifier
										lowerCamelCase__   =0
										for row_vector in self.array:
																for obj in row_vector:
																						lowerCamelCase__   =max(_lowerCamelCase      ,					len(str(_lowerCamelCase ) ) )
										lowerCamelCase__   =F'''%{max_element_length}s'''

										# Make string and return
										def single_line(_lowerCamelCase ) -> str:
																nonlocal string_format_identifier
																lowerCamelCase__   ="["
																line += ", ".join(string_format_identifier % (obj,) for obj in row_vector )
																line += "]"
																return line

										s += "\n".join(single_line(_lowerCamelCase ) for row_vector in self.array )
										return s

				def __repr__( self ):
										return str(self )

				def 						_a      ( self      ,					_lowerCamelCase ):
										if not (isinstance(_lowerCamelCase      ,					(list, tuple) ) and len(_lowerCamelCase ) == 2):
																return False
										elif not (0 <= loc[0] < self.row and 0 <= loc[1] < self.column):
																return False
										else:
																return True

				def __getitem__( self      ,					_lowerCamelCase ):
										assert self.validate_indicies(_lowerCamelCase )
										return self.array[loc[0]][loc[1]]

				def __setitem__( self      ,					_lowerCamelCase      ,					_lowerCamelCase ):
										assert self.validate_indicies(_lowerCamelCase )
										lowerCamelCase__   =value

				def __add__( self      ,					_lowerCamelCase ):
										assert isinstance(_lowerCamelCase      ,					_lowerCamelCase )
										assert self.row == another.row and self.column == another.column

										# Add
										lowerCamelCase__   =Matrix(self.row      ,					self.column )
										for r in range(self.row ):
																for c in range(self.column ):
																						lowerCamelCase__   =self[r, c] + another[r, c]
										return result

				def __neg__( self ):
										lowerCamelCase__   =Matrix(self.row      ,					self.column )
										for r in range(self.row ):
																for c in range(self.column ):
																						lowerCamelCase__   =-self[r, c]
										return result

				def __sub__( self      ,					_lowerCamelCase ):
										return self + (-another)

				def __mul__( self      ,					_lowerCamelCase ):

										if isinstance(_lowerCamelCase      ,					(int, float) ):  # Scalar multiplication
																lowerCamelCase__   =Matrix(self.row      ,					self.column )
																for r in range(self.row ):
																						for c in range(self.column ):
																												lowerCamelCase__   =self[r, c] * another
																return result
										elif isinstance(_lowerCamelCase      ,					_lowerCamelCase ):  # Matrix multiplication
																assert self.column == another.row
																lowerCamelCase__   =Matrix(self.row      ,					another.column )
																for r in range(self.row ):
																						for c in range(another.column ):
																												for i in range(self.column ):
																																		result[r, c] += self[r, i] * another[i, c]
																return result
										else:
																lowerCamelCase__   =F'''Unsupported type given for another ({type(_lowerCamelCase )})'''
																raise TypeError(_lowerCamelCase )

				def 						_a      ( self ):
										lowerCamelCase__   =Matrix(self.column      ,					self.row )
										for r in range(self.row ):
																for c in range(self.column ):
																						lowerCamelCase__   =self[r, c]
										return result






				def 						_a      ( self      ,					_lowerCamelCase      ,					_lowerCamelCase ):
										assert isinstance(_lowerCamelCase      ,					_lowerCamelCase ) and isinstance(_lowerCamelCase      ,					_lowerCamelCase )
										assert self.row == self.column == u.row == v.row  # u, v should be column vector
										assert u.column == v.column == 1  # u, v should be column vector

										# Calculate
										lowerCamelCase__   =v.transpose()
										lowerCamelCase__   =(v_t * self * u)[0, 0] + 1
										if numerator_factor == 0:
																return None  # It's not invertable
										return self - ((self * u) * (v_t * self) * (1.0 / numerator_factor))


# Testing
if __name__ == "__main__":





	def      lowerCamelCase_					(   )				->   None:




							'''simple docstring'''



							lowerCamelCase__   =Matrix(3  ,       3  ,       0							)
							for i in range(3							):
													lowerCamelCase__   =1
							print(F'''a^(-1) is {ainv}'''							)
							# u, v
							lowerCamelCase__   =Matrix(3  ,       1  ,       0							)
							lowerCamelCase__   , lowerCamelCase__   , lowerCamelCase__   =1, 2, -3
							lowerCamelCase__   =Matrix(3  ,       1  ,       0							)
							lowerCamelCase__   , lowerCamelCase__   , lowerCamelCase__   =4, -2, 5
							print(F'''u is {u}'''							)
							print(F'''v is {v}'''							)
							print(F'''uv^T is {u * v.transpose()}'''							)
							# Sherman Morrison
							print(F'''(a + uv^T)^(-1) is {ainv.sherman_morrison(__lowerCAmelCase  ,       __lowerCAmelCase							)}'''							)





	def      lowerCamelCase_					(   )				->   None:




							'''simple docstring'''



							import doctest

							doctest.testmod()





	testa()


